If one were to run this game repeatedly, the number of right and wrong guesses would be equal even though the group as a whole would win the money six out of eight times or 75 percent of the time. That is, half of all individual guesses are wrong, but three-fourths of the group responses are right!
Generalizations to situations with more than three people exist, but all solutions depend on finding a strategy that most of the time results in no one being wrong and every once in a while has everyone being wrong. With seven people playing, a strategy can be devised that wins the money 7/8ths of the time, with 15 players, 15/16ths of the time.
Are there other situations, say in the stock market, where independent pieces of information are provided to members of a group who can easily become aware of others' information but not of their own and, hence, where such strategies might work?
The idea behind these strategies can be phrased in terms of error-correcting codes, which are used in compact discs, modems, cell phones, and a host of other electronic devices. Not unrelated is the digit at the end of a bar code, a check digit which is (the units digit in) the sum of all the previous digit in the bar code.
P.S. Use As a Hoax?
Finally, it occurs to me that, were I so inclined, I could exploit the puzzle to appeal to gullible people desperate to find "evidence" for psychic phenomena.
After muttering a few incomprehensible New Age platitudes, I could describe the outcomes as a result of the three people attempting to mentally transmit hat colors and could further claim that whoever receives a strong enough signal from the others will speak up. Then I could observe that 3/4ths of the groups respond correctly and attribute this to their telepathic abilities.
Furthermore, by instructing my three co-conspirators not to follow the guessing rules strictly, I could further muddy the waters and inspire speculation about why people more often respond correctly when both of the other two are wearing the same color hat.
Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on ABCNEWS.com appears every month.